On monodromy eigenfunctions of Heun equations and boundaries of phase-lock areas in a model of overdamped Josephson effect

Abstract

We study a family of double confluent Heun equations of the form $\mathcal E=0$, where $\mathcal L=\mathcal L_{\lambda,\mu,n}$ is a family of differential operators of order two. They depend on complex parameters $\lambda$, $\mu$, $n$. Its restriction to real parameter domain $\lambda+\mu^2>0$ is a linearization of the family of nonlinear equations on two-torus modeling the Josephson effect in superconductivity. We show that for every $b,n\in\mathbb C$ satisfying a certain "non-resonance condition" and every $\lambda,\mu\in\mathbb C$, $\mu\neq0$ there exists an entire function $f_{\pm}:\mathbb C\to\mathbb C$ (unique up to constant factor) such that $z^{-b}\mathcal L(z^b f_{\pm}(z^{\pm1}))=d_{0\pm}+d_{1\pm}z$ for some $d_{0\pm},d_{1\pm}\in\mathbb C$. The constants $d_{j,\pm}$ are expressed as functions of the parameters. This result has several applications. First of all, it gives the description of those $\lambda$, $\mu$, $n$, $b$ for which the monodromy of the Heun equation has eigenvalue $e^{2\pi i b}$. It also describes those $\lambda$, $\mu$, $n$ for which the monodromy is parabolic: has multiple eigenvalue. We consider the rotation number $\rho$ of the dynamical system on two-torus as a function of parameters restricted to a surface $\lambda+\mu^2=const$. The phase-lock areas are its level sets having non-empty interiors. For general families of dynamical systems the problem to describe the boundaries of the phase-lock areas is known to be very complicated. Here we include the results in this direction obtained by methods of complex variables. In our case the phase-lock areas exist only for integer rotation numbers (quantization effect). The result on parabolic monodromy implies the description of the union of their boundaries by an explicit functional equation. For every $\theta\notin\mathbb Z$ we get a description of the set $\{\rho\equiv\pm\theta(mod2\mathbb Z)\}$.